While Kolmogorov's definition of probability looks a bit spooky at first, I definitely recommend going through it at some point. Not only it feels nice, it also makes understanding probabilistic concepts easier (e.g. understanding that a statistical space is simply a "lifted", or rather parametrized, version of a probabilistic space).
I like the stories about Warren Buffett's fascination with the dice:
> Buffett once attempted to win a game of dice with Bill Gates using nontransitive dice. "Buffett suggested that each of them choose one of the dice, then discard the other two. They would bet on who would roll the highest number most often. Buffett offered to let Gates pick his die first. This suggestion instantly aroused Gates's curiosity. He asked to examine the dice, after which he demanded that Buffett choose first."[1]
> In 2010, Wall Street Journal magazine quoted Sharon Osberg, Buffett's bridge partner, saying that when she first visited his office 20 years earlier, he tricked her into playing a game with nontransitive dice that could not be won and "thought it was hilarious".[2]
this isnt paradoxical at all, it's just you are looking at only 2 dice at a time not the full joint probability of a < b < c. i can come up with similarly retarded "paradoxical conclusions" if i took enough samples of different tiny slices of things.
dude is just trying to sound smart with yet another "counterintuitive probability post"
Imagine a tournament to choose the best tennis player. If you imagine these dice as a simple model of a tennis player, there can be no best tennis player. So tennis tournaments make no sense. This is paradoxical if you believe in tennis rankings.
> This is paradoxical if you believe in tennis rankings.
So the solution is to go the rational way and stop believing in ranking if you claim that tennis ranking is based on a mathematical model that has the non-transitivity property. Where exactly is the problem?
It makes sense to me that the best athlete cannot be determined in some scenarios, or that the term "best" is simply meaningless. Due to training and/or genetics, player A might be strong in areas where player B is weak. Likewise, player B might be strong in areas where player C is weak.
Neither of these statements gives us any insight into what will happen if player A faces player C, but a traditional tournament structure might well keep A and C from ever facing each other.
A more revealing model might be a race, since one's performance is not obviously linked to another's and there's only one variable being measured (time to completion).
Not even true in that case, because races often come down to luck with pit stops and how aggressive other drivers are in protecting their space. If you can't pass the dude without rear ending him, you're not going to pass him, doesn't matter how better your "single-racer" driving skills or faster your car is on paper.
Even in footraces, the competitors still have influence on each other with drafting and late starting and whatnot.
Uhh, what if I told you that like most people with common sense and who realize that the initial conditions of a tournament are fixed in advance and that the ranking at the end only applies for that tournament and there are many other factors that influence who wins and loses in each match, I take tennis rankings with a grain of salt?
Hell I can even accept that player A > B > C > A if each has different skills and specialities that they bring to the table, without trapping myself and others into declaring it's some "paradoxical nonsense". Do we say rock < paper < scissors < rock and declare it's the end of mathematics?
Does anyone anywhere say that if something may appear to some people paradoxical (have you ever thought about what paradoxical means?) it is the end of mathematics?
> Examples of binary relations are binary operations (like +,–,⋅,/) on N,Z,Q,R,C,Rn, … etc. (where defined), since functions are special cases of relations, or, the order relations <,≤,>,≥ on R, for instance.
A function of two variables is a ternary relation, not a binary relation. /pedant
This is completely incorrect. There are two inputs and one output, e.g., ℝ⨯ℝ→ℝ. Since there are two inputs, it's a binary operator. Since there are three inputs and outputs combined, it's a ternary relation.
Likewise, a unary operator is a binary relation. For example, x ↦ -x, or x ↦ x+1.
Idk what to tell you other than that you should check the definitions. A unary function is a special case of a binary relation and in general an n-ary function is a special case of a n+1-ary relation.
Edit - Maybe this clarifies it: When we talk about a function we designate one of its variables as the 'output' and talk about its arity in terms of the inputs. When we talk about a relation there is no distinguished output: A relation is a fully symmetrical definition. That's why the arity of a function is one less than the arity of that same function thought of as a relation between its inputs and output.
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[ 5.2 ms ] story [ 65.5 ms ] thread> Buffett once attempted to win a game of dice with Bill Gates using nontransitive dice. "Buffett suggested that each of them choose one of the dice, then discard the other two. They would bet on who would roll the highest number most often. Buffett offered to let Gates pick his die first. This suggestion instantly aroused Gates's curiosity. He asked to examine the dice, after which he demanded that Buffett choose first."[1]
> In 2010, Wall Street Journal magazine quoted Sharon Osberg, Buffett's bridge partner, saying that when she first visited his office 20 years earlier, he tricked her into playing a game with nontransitive dice that could not be won and "thought it was hilarious".[2]
dude is just trying to sound smart with yet another "counterintuitive probability post"
Federer > Djokovic (well not lately)
Djokovic > Nadal
Nadal > Federer
Everything depends on the draw. Same with the initial choice of dice.
So the solution is to go the rational way and stop believing in ranking if you claim that tennis ranking is based on a mathematical model that has the non-transitivity property. Where exactly is the problem?
Neither of these statements gives us any insight into what will happen if player A faces player C, but a traditional tournament structure might well keep A and C from ever facing each other.
Even in footraces, the competitors still have influence on each other with drafting and late starting and whatnot.
Hell I can even accept that player A > B > C > A if each has different skills and specialities that they bring to the table, without trapping myself and others into declaring it's some "paradoxical nonsense". Do we say rock < paper < scissors < rock and declare it's the end of mathematics?
A function of two variables is a ternary relation, not a binary relation. /pedant
Likewise, a unary operator is a binary relation. For example, x ↦ -x, or x ↦ x+1.
Edit - Maybe this clarifies it: When we talk about a function we designate one of its variables as the 'output' and talk about its arity in terms of the inputs. When we talk about a relation there is no distinguished output: A relation is a fully symmetrical definition. That's why the arity of a function is one less than the arity of that same function thought of as a relation between its inputs and output.